Ghost

You completed all missions after the prologue, alerting or killing no one but key targets

For enemies you cannot kill (or let come to harm by) see Clean Hands. Alerts only count if an enemy fills up its alert meter (the option to see the alert meter is in the options; turn it on). All enemies who have the alert meter will count towards this achievement / trophy.

To avoid being spotted by her, you may want to downshift the game's difficulty to Easy first (this should not affect the achievement or trophy, since all the conditions are difficulty independent). Downgrading the enemy AI to easy severely impacts their visibility range.

Before placing the cameo into the furnace (a point of no return), take a look around.

There is a steel bar holding assorted cooking pots that connects to a ledge running around the furnace in her lair (the furnace where you incinerate her cameo); as soon as you place her cameo in the furnace, Blink to the bar (or try to squeeze Corvo onto the ledge) while Granny rants and raves. Corvo should "stealth" himself there, and be reasonably undetectable (you may want to shuffle to the corner near the door to the Outsider Shrine to be sure).

Granny, after delivering her crazy old woman diatribe, should vanish and reappear outside the furnace room. Once she does, you can use the furnace switch to destroy the cameo from the safety of the ledge (or the steel bar).

From that point on, you simply need to Blink to a spot that's free of rats and shoot a sleep dart in Granny's octogenarian ass to make her take a nap. Dark Vision helps immensely in seeing where Granny teleports to after each rat swarm summoning.